Division Algebras of Prime Degree and Maximal Galois $p$-Extensions

http://dx.doi.org/10.4153/CJM-2007-028-7
Canad. J. Math. 59(2007), 658-672

  Published:2007-06-01
 Printed: Jun 2007

Let $p$ be an odd prime number, and let $F$ be a field of characteristic not $p$ and not containing the group $\mu_p$ of $p$-th roots of unity. We consider cyclic $p$-algebras over $F$ by descent from $L = F(\mu_p)$. We generalize a theorem of Albert by showing that if $\mu_{p^n} \subseteq L$, then a division algebra $D$ of degree $p^n$ over $F$ is a cyclic algebra if and only if there is $d\in D$ with $d^{p^n}\in F - F^p$. Let $F(p)$ be the maximal $p$-extension of $F$. We show that $F(p)$ has a noncyclic algebra of degree $p$ if and only if a certain eigencomponent of the $p$-torsion of $\Br(F(p)(\mu_p))$ is nontrivial. To get a better understanding of $F(p)$, we consider the valuations on $F(p)$ with residue characteristic not $p$, and determine what residue fields and value groups can occur. Our results support the conjecture that the $p$ torsion in $\Br(F(p))$ is always trivial.